Tall, Dark, and Mysterious


One man, n votes

File under: Character Writ Large, Queen of Sciences. Posted by Moebius Stripper at 1:20 pm.

Buoyed by a surge of student queries of the form, “what applications does this have in real life?” I was inspired to dust off an old copy - the library’s - of John Allen Paulos’ A Mathematician Reads the Newspaper. By the time I finished rereading the brief aside on measuring shareholder and voter power, I had abandoned my original goal of making precalculus relevant to my pupils, and was wondering if there were any analyses online about the amounts of power held by the various states in the electoral college that went beyond the standard “wooo, gotta worry about Florida”-type punditry. Naturally, there were, and since this is about the only aspect of the US election that I can think about without wanting to claw my eyes out, I thought I’d post some of them.

To the best of my knowledge, the Banzhaf index is the standard means of measuring power of groups in block voting systems, such as the electoral college, in which each state’s vote is weighted. The Banzhaf power index for Florida, for example, is computed by considering all the state-by-state possible outcomes in the election - one outcome being the possibility that California’s 54 electoral votes go to Kerry, New York’s 33 go to Kerry, Texas’ 32 go to Bush… - and then counting the numbers of those outcomes that are swung by Florida. Here is a state-by-state list of the Banzhaf power indices for the 50 [thanks, Chris] states and DC. (The power indices in the other columns are also defined.) Florida, the largest swing state, has a power index of 0.193864, meaning that in 19.4% of possible outcomes, neither Bush nor Kerry will have enough electoral votes to win the presidency before Florida is counted. Compare this figure to the relatively small ~4.6% of total electoral votes allocated to Florida. (California, meanwhile, is critical in nearly half of all possible outcomes.) The runtime of the programs doing these computations is already pretty high (O(2^n )), but I wonder if there are any probabilistic variations on this index as applied to the electoral college. In the standard computation, for instance, an outcome that gives California’s 54 votes to Bush and Texas’ 32 to Kerry is weighted the same as the far more likely alternative. A friend of mine from Mathcamp has written some Maple routines evaluating different power indices; someone who keeps up with US politics better than I do could probably make the modification pretty easily.

Going a bit further: I haven’t yet read this detailed article about the Banzhaf power index, but it also contains an analysis of how much power each individual voter has - taking into consideration the population of the states as well as their voice in the electoral college. Despite California’s large population, its voters have the most say - each is 3.34 times as powerful as a single Montana voter. (This, I presume, makes certain assumptions - for instance, that the percentage of registered voters who actually show up is constant from state to state.)

Paulos gives a simple and dramatic example of the relative usefulness of the Banzhaf index versus more standard measurements: consider a company with three shareholders, who respectively own 49%, 35%, and 16% of the company. Although the first’s share is more than triple the third’s, all have equal voting power: in a yes/no vote, whichever side attracts at least two of the voters, carries. Consequently, all shareholders have the same Banzhaf power index - in this case, 1/2. On the other hand, if they held 51%, 35%, and 18% respectively, the first shareholder’s vote is clearly the only one that matters. His power index is 1, and the others’ are each 0.


  1. I did the electoral college Banzhaf thing, programming it in Fortran, in 1992… and I couldn’t get past the election of about 1824 or so. Too many states. I figured out my next jump up would take about 1000 years to run. That was actually running through the 2^n possibilities.

    However, there are people who have come up with Monte Carlo routines to make estimates. And then, some people have done analyses taking into consideration that certain states are “locks” for particular parties, and thus their power goes to zero, effectively. Of course, some of those “lock” states aren’t always a sure thing, but generally in one election cycle you know where they stand. For example, no one comes campaigning for President in New York City other than to raise money — I’ve been getting political calls, but they’re all solicitations for donations… not even a basic Get Out the Vote exhortation. At least I don’t live in Ohio. I feel sorry for those guys.

    - meep — 10/26/2004 @ 11:28 pm

  2. Unless they added one while I wasn’t looking, I think you mean 50 states plus D.C.

    - Chris — 10/27/2004 @ 2:09 am

  3. Chris - oops, thanks - I’ve corrected it.

    - Moebius Stripper — 10/27/2004 @ 8:15 am

  4. Taking a step backwards from the particular mathematics (and, parenthetically, all this stuff about Order 2^n and other computationally intensive things reminds me of the traveling salesman problem), I would like to add that the differing relative power of a vote from state to state points up what in my mind is an inherent unfairness to the electoral college system. Wouldn’t it be so much fairer and better, not to mention simpler, simply to elect the president by popular vote, such that whoever gets the most votes wins? Yeah, I should keep on dreaming.

    - wes — 10/27/2004 @ 9:24 am

  5. I’m not too familiar with the electoral college - do the electoral votes actually translate into actual human beings representing the state? That’s pretty much the way Canada’s parliamentary system works - if a plurality of people in my riding, say, vote Liberal, then that riding sends a Liberal Member of Parliament to Ottawa. With such a system we get things like, say, a party getting 20% of the popular vote and 2% of the seats, which was what happened in 1993 to the Progressive Conservative Party.

    - Moebius Stripper — 10/27/2004 @ 11:25 am

  6. Yes, MS. A vote for Bush translates into a selection of people who have pledged to vote for Bush at the electoral college convention. The exact details vary from state to state. In particular, my understanding is that some states allow for the splitting of their electoral college votes in certain circumstances (I am not sure about the details here).
    There have also been some cases of delegates breaking with their pledges — most recently in one of the ’80s elections, a delegate made a protest vote in an attempt to spark reform of the system.

    Apparently, the historical motivation was for citizens to choose representatives whose judgement they trusted. These representatives would then discuss who the president should be. There is NO constitutional requirement for the electors to be chosen by vote. The state legislature could choose them arbitrarily, if they made that the state law.

    A huge difference between the Canadian parliamentary system and the American presidential election is that the discrete chunks in the US are much larger.

    I believe a motivation for having ridings w/ specific representatives in Canada, say, is that many issues are region/community specific. The concerns and priorities of a rural riding in Alberta are likely to be very different from the concerns and priorities of a downtown Toronto riding.

    In the States, for the presidential elections, that argument breaks down. A voter in Chicago has more in common, probably, with a voter in New York City than either have with their rural down/up state neighbours.

    Wes, I think a problem with what you propose is that one-man-one-vote would put too much power with major urban centres. Of course, currently, it seems the rural states in the US have disproportionate power relative to cities. Some middle course would seem fairest to me. Perhaps half of the electoral college based on total popular vote and then the rest by congressional district?

    - Sam — 10/27/2004 @ 8:17 pm

  7. A more general problem with electoral college vs. popular vote vs. parliamentary ridings vs. anything else is that none of the solutions are ever fair, they only distribute the unfairness differently.
    That of course doesn’t mean that there’s a better distribution of unfairness than the current system, but it’s usually a lot easier to say what’s wrong with the current system than to come up with a better one.

    (My own pet idea for Canadian electoral reform is to merge similar and nearby ridings and elect multiple candidates at-large in those larger ridings. So a city with, say, three ridings under the current system would end up being a single riding that elects three MPs, and large rural areas could be merged into a small number of ridings with multiple MPs each. The idea behind this is that the makeup of parliament gives a better representation of popular vote between the major parties (f’rexample, five ridings with a small preference for the Liberals over whatever the conservatives are called this week currently elect five Liberal MPs; as a single riding with five seats they’d elect three Liberals and two conservatives), and gives independent and minor-party candidates a chance at actually getting seats in large enough numbers to have nonzero influence (a small party could run a single candidate in a riding, while larger parties would run as many as they thought they could get seats). This could easily have the side effect of making minority governments common enough that politicians would have to learn how to deal with them properly. I have yet to find somebody who’s interested enough in politics to argue about this in enough depth to come up with any of the problems with it, though.)

    - dave — 10/27/2004 @ 10:02 pm

  8. Where, if I’m not mistaken, 18% = 14% :)

    Wes, check out Kenny’s commentary on the electoral college at Cardinal Collective. He has some interesting ideas about better systems, and pretty good reasons to oppose direct ellection.

    - Theo — 10/29/2004 @ 10:57 pm

  9. Actually, I remember noticing specifically a week or so before the election that Sam Wang at Princeton, in his probabilistic electoral vote calculator, basically calculated the probabilized Banzhaf index of each state. Using the latest polls, he got a probability that each state would go each way (this is probably the weakest point in the system, because I think it was more a measure of how likely a poll was to capture the “actual result”, rather than a measure of how many times the election would go one way or the other if it was run a bunch of times). At a point further down in the page, he showed a table with about 20 states, and showed the probability of each of the four events, where the state goes to Kerry and Kerry wins, where it goes to Kerry and Bush wins, where it goes to Bush and Kerry wins, and where it goes to Bush and Bush wins. The probability that California went to the winner of the election was about 55 or something at some point that I looked, while the probability that Iowa did was about 70, and the probability that Ohio did was about 90.

    Allocating parliament and congress in district-based manners makes perfect sense for the reasons above - that for every regional community, there should be someone in government who is beholden to that community. For presidential elections of course, the argument is much more tenuous. Because direct popular vote elections would become totally city-centric, and the current system creates a collection of “swing states”, I would recommend a change whereby each state allocates its electors proportionally, rather than winner-take-all. (Right now, Maine and Nebraska allocate them by congressional district, but I think that’s a bad idea because gerrymandering is a big enough problem for congress without it affecting the president as well.) One “problem” with proportional allocation is that third-party candidates would get a large number of electors, and thus no one would be able to get a majority in the actual election, but this can be settled either by giving the electors a runoff, or allowing them to come to whatever agreement suits a majority.

    - Kenny — 11/10/2004 @ 1:01 am

RSS feed for comments on this post.

Sorry, the comment form is closed at this time.